Problem: Let $\mathbf{a}$ and $\mathbf{b}$ be two vectors such that
\[\|\mathbf{a} + \mathbf{b}\| = \|\mathbf{b}\|.\]Find the angle between the vectors $\mathbf{a} + 2 \mathbf{b}$ and $\mathbf{a},$ in degrees
Solution: From the equation $\|\mathbf{a} + \mathbf{b}\| = \|\mathbf{b}\|,$ $\|\mathbf{a} + \mathbf{b}\|^2 = \|\mathbf{b}\|^2,$ so
\[(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b}) = \mathbf{b} \cdot \mathbf{b}.\]Expanding, we get $\mathbf{a} \cdot \mathbf{a} + 2 \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{b},$ so
\[\mathbf{a} \cdot \mathbf{a} + 2 \mathbf{a} \cdot \mathbf{b} = 0.\]We can write this as $\mathbf{a} \cdot (\mathbf{a} + 2 \mathbf{b}) = 0.$  Thus, the vectors $\mathbf{a}$ and $\mathbf{a} + 2 \mathbf{b}$ are orthogonal, and the angle between them is $\boxed{90^\circ}.$